integration

412 INTEGRAL CALCULUS
= 1 [−ln(a − x) + ln (a + x)] + c
2a
= 1
2a ln ( a + x
a − x
Problem 12.
∫ 2
0
Evaluate
5
(9 − x 2 ) dx,
)
+ c
correct to 4 decimal places.
From Problem 11,
∫ 2
0
[ ( )]
5
1 3 + x 2
(9 − x 2 ) dx = 5 2(3) ln 3 − x 0
= 5 [ln 5 ]
6 1 − ln 1
= 1.3412, correct to 4
decimal places
Now try the following exercise.
Exercise 165 Further problems on integration
using partial fractions with quadratic
factors
∫
x 2 − x − 13
1. Determine
(x 2 + 7)(x − 2) dx
⎡
⎣ ln (x2 + 7) + √ 3 tan −1 x ⎤
√
7 7
⎦
− ln (x − 2) + c
In Problems 2 to 4, evaluate the definite integrals
correct to 4 significant figures.
2.
3.
∫ 6
5
∫ 2
1
∫ 5
6x − 5
(x − 4)(x 2 dx [0.5880]
+ 3)
4
(16 − x 2 dx [0.2939]
)
2
(x 2 − 9)
4.
dx [0.1865]
4
∫ 2
( 2 + θ + 6θ 2 − 2θ 3 )
5. Show that
θ 2 (θ 2 dθ
+ 1)
1
= 1.606, correct to 4 significant figures.